Download App

DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS5 Marks
Question
Using the proprties of determinants in Exercise:
$\begin{vmatrix}\text{y}+\text{z}&\text{z}&\text{y}\\\text{z}&\text{z}+\text{x}&\text{x}\\\text{y}&\text{x}&\text{x}+\text{y}\end{vmatrix}=4\text{xyz}$
✓

Answer

We have to prove,
$\begin{vmatrix}\text{y}+\text{z}&\text{z}&\text{y}\\\text{z}&\text{z}+\text{x}&\text{x}\\\text{y}&\text{x}&\text{x}+\text{y}\end{vmatrix}=4\text{xyz}$
$\therefore\ \text{L.H.S}=\begin{vmatrix}\text{y}+\text{z}&\text{z}&\text{y}\\\text{z}&\text{z}+\text{x}&\text{x}\\\text{y}&\text{x}&\text{x}+\text{y}\end{vmatrix}$
$=\begin{vmatrix}\text{y}+\text{z}+\text{z}+\text{y}&\text{z}&\text{y}\\\text{z}+\text{z}+\text{x}+\text{x}&\text{z}+\text{x}&\text{x}\\\text{y}+\text{x}+\text{x}+\text{y}&\text{x}&\text{x}+\text{y}\end{vmatrix}$ $\big[\because\text{C}_1\rightarrow\text{C}_1+\text{C}_2+\text{C}_3\big]$
$=2\begin{vmatrix}(\text{y}+\text{z})&\text{z}&\text{y}\$\text{z}+\text{x})&\text{z}+\text{x}&\text{x}\$\text{x}+\text{y})&\text{x}&\text{x}+\text{y}\end{vmatrix}$
$\big[\text{Taking 2 common from C}_1\big]$
$=2\begin{vmatrix}\text{y}&\text{z}&\text{y}\\0&\text{z}+\text{x}&\text{x}\\\text{y}&\text{x}&\text{x}+\text{y}\end{vmatrix}$ $[\because\ \text{C}_1\rightarrow\text{C}_1-\text{C}_2]$
$=2\begin{vmatrix}0&\text{z}-\text{x}&-\text{x}\\0&\text{z}+\text{x}&\text{x}\\\text{y}&\text{x}&\text{x}+\text{y}\end{vmatrix}$ $[\because\ \text{R}_1\rightarrow\text{R}_1-\text{R}_3]$
$=2\big[\text{y}(\text{xz}-\text{x}^2+\text{xz}+\text{x}^2)\big]$
$=4\text{xyz}=\text{R.H.S}$
Hence proved.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from DETERMINANTS→DETERMINANTS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the differential equation of the family of curve $\text{y}=\text{Ae}^\text{2x}+\text{Be}^{-2\text{x}},$ where A and B are arbitrary constants.
→
An article manufactured by a company consists of two parts X and Y. In the process of manufacture of the part X, 9 out of 100 parts may be defective. Similarly, 5 out of 100 are likely to be defective in the manufacture of part Y. Calculate the probability that the assembled product will not be defective.
→
If $\vec{\text{a}}=2\hat{\text{i}}+5\hat{\text{j}}-7\hat{\text{k}},\vec{\text{b}}=-3\hat{\text{i}}+4\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{c}}=\hat{\text{i}}-2\hat{\text{j}}-3\hat{\text{k}},$ compute $\big(\vec{\text{a}}\times\vec{\text{b}}\big)\times\vec{\text{c}}$ and $\vec{\text{a}}\times\big(\vec{\text{b}}\times\vec{\text{c}}\big)$ and verify that these are not equal.
→
Solve the following system of equations by matrix method:
$3x + y = 19$
$3x - y = 23$
→
Solve the following differential equations:$\tan\text{y}\frac{\text{dy}}{\text{dx}}=\sin(\text{x}+\text{y})+\sin(\text{x}-\text{y})$
→
Show that the lines $\frac{\text{x}+1}{3}=\frac{\text{y}+3}{5}=\frac{\text{z}+5}{7}$ and $\frac{\text{x}-2}{1}=\frac{\text{y}-4}{3}=\frac{\text{z}-6}{5}$ intersect. Find their point of intersection.
→
Find the maximum slope of the curve $y = −x^3+ 3x^2+ 2x − 27.$
→
Solve the following differential equation:
x dy – y dx = $\sqrt{\text{x}^{2}+\text{y}^{2}}\text{ dx}$ .
→
$\text{Find} \frac{\text{dy}}{\text{dx}} \text{if (x}^{2} + \text{y}^2)^{2} = \text{xy.}$
→
$\text{if } \vec{\text{a}} = 2\hat{\text{i}} + \hat{\text{j}} - \hat{\text{k}}, \vec{\text{b}} = 4\hat{\text{i}} - 7\hat{\text{j}} + \hat{\text{k}}, \text{find a vector } \vec{\text{c}} \text{ such that } \vec{\text{a}} \times \vec{\text{c}} \text{ and } \vec{\text{a }} . \vec{\text{c}} = 6.$
→